220 SCIENCE IN GRECO-ROMAN ANTIQUITY 



nature tends to be penetrated by rationality, but this 

 penetration is never complete because of the resistance 

 which matter offers to form, and this is why individual 

 beings are always imperfect examples of form. The 

 numerical and spatial relations as conceived by arith- 

 metic and geometry have a totally different character, 

 for these relations are eternal, independent of time, 

 of physical place and of circumstances. If, as Aristotle 

 thought, mathematical entities have been gradually 

 disentangled by abstraction from the sensible world, 

 having once been obtained by this process, they appear 

 in a perfect and immutable form. This being so, 

 individual mathematical entities are an exact reproduc- 

 tion of the genus and species to which they belong. 

 Every isosceles triangle, whether small or large, 

 possesses completely and perfectly all the properties 

 of the isosceles triangle, in this sense, that, having two 

 sides equal, it necessarily has two angles equal. Mathe- 

 matical entities attain their perfect form without any 

 progression in time. The abstract relation of which 

 they are constituted is eternal, or rather it is a relation, 

 irrespective of time, between laws and consequences, 

 in which the efficient cause and the final cause are 

 merged by an indivisible action of the mind. 



This fact determines the nature of mathematical 

 conceptions and demonstrations within the following 

 limits : 



The primary propositions (axioms, definitions, postu- 

 lates) must avoid making any appeal to obscure ideas 

 of the sensible intuition such as indefinite dichotomous 

 divisibility and the relation of motion to space. 



On the other hand, in geometrical demonstration it 

 is most necessary to use static methods, and to consider 

 as foreign to science the constructions which result 

 from the meeting of two lines in motion. 



In the same way, in dealing with integration, the 

 passage to the limit cannot be directly effected. It can 



